Group

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Group
4 C4 t: f& h( z- iA group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
; E8 K$ W- x$ Q- U6 { to form well-defined products and which furthermore satisfy the following conditions:
  j1 L' O" R4 B1. Closure: If and are two elements in , then the product is also in .
2. Associativity: The defined multiplication is associative, i.e., for all , .
1 U$ I& [3 E! S! N3. Identity: There is an Identity Element
(a.k.a. , , or ) such that for every element .
4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
A group is therefore a Monoid
$ q' E) r! y' ~4 \1 @5 o for which every element is invertible. A group must contain at least one element.

9 H/ A8 J# T" z0 l2 J! ~7 B" }The study of groups is known as Group Theory
. If there are a finite number of elements, the group is called a Finite Group
and the number of elements is called the Order
of the group.

Since each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1)

must also be a member. Now apply to ,
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+ N5 J$ m' X2 W3 W

(2) ( B) N0 n1 O, P7 R4 `

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" U, `' |, @& OBut
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	# ~5 v' d8 l; s" ?. ^3 l, }
7 E- |- J# E- p0 v' p& y
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so

(4) ; f/ H' k* u6 F) H( q& t

. R- B. I" Y( W3 r3 V  d
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which means that
9 G  z" J, x/ @. T

(5)

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  ~# Z! [, V; i. m1 uand
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(6)

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OLS

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